The Technical Details
tab on the product page has some of this information. There is also a tutorial page
which describes a little more about our character LCDs, including this one.
There is a single RGB LED for the backlight. There are four separate lines (R, G, B, power) for the backlight. The LCD has resistors on board already so that you can drive it with 5V logic and the current draw will be ~20mA per LED. I don't know the specific forward voltages of the LED segments, but RGBs are typically around 2V. With that assumption, the on-board resistors will probably give you around 10mA current draw on 3.6V
You can PWM the RGB control lines using inverse logic (i.e., low = 'ON', high = 'OFF').
You will need 6 digital lines to control the display.